The　saturated　boundary　stabilization　problem　for　quasi-linear　hyperbolic　systems　of　balance　laws　is　considered　under　H2-norm　in　this　paper,　where　the　boundary　conditions　of　the　system　are　subject　to　actuator　saturations.　The　resulting　closed-loop　system　is　proven　to　be　locally　exponentially　stable　with　respect　to　the　steady　states　in　the　presence　of　saturations.　To　this　end,　the　sector　nonlinearity　model　is　introduced　to　deal　with　the　saturation　term,　and　then　sufficient　conditions　for　ensuring　the　locally　exponential　stability　are　established　in　terms　of　a　set　of　matrix　inequalities　by　employing　the　Lyapunov　function　method　along　with　a　sector　condition.　Furthermore,　these　results　are　applied　to　the　stabilization　of　the　two-lane　traffic　flow　dynamic　represented　by　Lighthill-Whitham-Richards　(LWR)　model.　By　utilizing　variable　speed　limit　(VSL)　devices,　a　saturated　boundary　feedback　controller　is　designed　to　stabilize　the　two-lane　traffic　flow,　and　the　exponential　convergence　of　the　quasi-linear　traffic　flow　system　in　H2　sense　is　validated　by　numerical　simulations.